Errata

Bruce Bartlett: I don’t understand the ss map in Example 5.1.11 in the online version of the book. It’s supposed to be the automorphism

(1)s:Γ1,1→Γ1,1
s : \Gamma_{1,1} \rightarrow \Gamma_{1,1}

of the torus with one boundary circle which is the analogue of a map which I do understand, the automorphism

(2)st:Γ1,0→Γ1,0
s_t : \Gamma_{1,0} \rightarrow \Gamma_{1,0}

of the ordinary torus (no boundary circle) to itself, which sends

(3)(θ,ϕ)↦(ϕ,−θ).
(\theta, \phi) \mapsto (\phi, -\theta).

But the map sts_t doesn’t seem to work if there is a boundary circle involved. Because an automorphism of Γ1,1\Gamma_{1,1} is supposed to be the identity on the boundary circle, but the map sts_t isn’t: it rotates the boundary circle a quarter rotation (right?). The reason I say so is the following picture: we think of the torus as ℝ2/ℤ2\mathbb{R}^2 / \mathbb{Z}^2, as Bakalov and Kirillov encourage us to do. Then the presence of the boundary circle can be thought of as having a little tangent vector pointing to the right at all the lattice positions (this uses their alternative Definition 5.1.10 the extended surface category). The map (x,y)↦(y,−x)(x,y) \mapsto (y, -x) rotates the plane by a quarter rotation and takes the lattice to itself, hence it descends to the quotient. But this map rotates the tangent vector by a quarter rotation, and we’re supposed to fix the tangent vectors!

So how does the map ss work?

Answer: Domenico Fiorenza explained it as follows: consider the torus with one boundary circle as a square made of elastic material with with opposite sides identified, with an open disc missing in the center. Then turn the square through one quarter of a rotation, while keeping the boundary circle fixed. That’s the ss map.

If you picture it at the level of the plane (i.e. before we mod R2R^2 out by Z2Z^2), then you need to imagine a whole bunch of circles (“knobs”) located at the integer lattice points of the plane. Get a whole bunch of friends (one for each knob), and then count down “3,2,1,turn!” and turn the knobs through a quarter revolution clockwise (this distorts the elastic material from which the plane is made). Then rotate the whole thing rigidly one quarter of a revolution counterclockwise. The resultant map is the ss-map at the level of the plane.